1. ## Proving...

Prove that if (v1, ... , vn) is a basis of V, then so is(v1, v2-v1, ... , vn - vn-1).

I made the initial assumption that if V remained the same number that by subtracting v1, youd always end up with v1, and so on. But obviously I came to the conclusion that this would have to be wrong, because the set of (v1, .. , vn) could represent any numbers. So I have no idea how to tackle this one, any help would be greatly appreciated. Thanks.

2. If $\displaystyle a_i$ are such that $\displaystyle \sum_{i=1} ^{n} \ a_iv_i =0$ then $\displaystyle a_i =0$ for all $\displaystyle i$. Now suppose there are scalars $\displaystyle b_i$ such that $\displaystyle \sum_{i=1} ^{n} \ b_i(v_i-v_{i-1})=0$ where $\displaystyle v_0=0$ then we get $\displaystyle \sum_{i=1} ^{n} \ (b_i - b_{i+1})v_i=0$ where $\displaystyle b_{n+1}=0$ then $\displaystyle b_n=0$, and so $\displaystyle b_{n-1}=0$ ... $\displaystyle b_1=0$. Which means they're l.i. and therefore a basis.

3. Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?

4. Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
(This is a std theorm and can be found in most of the texts on linear algebra)

5. Originally Posted by aman_cc
Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
(This is a std theorm and can be found in most of the texts on linear algebra)
Yes I understand that it is a standard theorem, just as we can say X²=4 when x=2. But that is not what I am asking, I am asking how to PROVE this theorem. That is a completely different method. However I appreciate your input.

6. Originally Posted by GreenDay14
Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?
What is your definition of basis?

7. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V.
You are given that the first n vectors form a basis for V so you know that V has dimension n. You are given another n vectors in V. To show that they are a basis for V you need only show that they span V or that they are independent.